Bonsean inequality

The Bonsian inequality is a theorem about the growth of prime numbers . It says that the square of a prime number is smaller than the product of the smaller prime numbers. The inequality was found and published by the Münster student Bonse in 1907. It was made accessible to a wide audience through the popular scientific mathematics book Von Numbers undfiguren by the two mathematicians Hans Rademacher (1892-1969) and Otto Toeplitz (1881-1940).

Formal: If the sequence of prime numbers denotes, then applies to all : ${\displaystyle (p_{n})_{n=1,\ldots }}$ ${\displaystyle n\geq 5}$

${\displaystyle {p_{n}}^{2}<p_{1}\cdot p_{2}\cdot p_{3}\cdots p_{n-1}=2\cdot 3\cdot 5\cdots p_{n-1}}$.

For not apply this inequality. So it is ${\displaystyle n\leq 4}$

${\displaystyle {4=2^{2}>1}}$

${\displaystyle {9=3^{2}>2=2}}$

${\displaystyle {25=5^{2}>2\cdot 3=6}}$

${\displaystyle {49=7^{2}>2\cdot 3\cdot 5=30}}$

---

${\displaystyle {121=11^{2}<2\cdot 3\cdot 5\cdot 7=210}}$

${\displaystyle {169=13^{2}<2\cdot 3\cdot 5\cdot 7\cdot 11=2310}}$

${\displaystyle {289=17^{2}<2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13=30030}}$

etc.

Tightening

As Rademacher and Toeplitz note, there are better results than Bonse's inequality; such as an inequality found by Pafnuti Lwowitsch Chebyshev , which states that each prime number is less than twice the respective preceding prime number. But these better results can only be proven with powerful means of higher mathematics, while Bonse only needed elementary means to prove his inequality.

An even more severe constraint even predicts a prime number between two square numbers. This is known as the Legendre's Conjecture , but it has not yet been proven.

Math applications

In 2007 Robert J. Betts described how one can get statements about the size of prime number gaps with the help of Bonsian inequality, which are not as strong as other known estimates, but can be derived in a simpler way.

Individual evidence

1. ↑ Hans Rademacher, Otto Toeplitz: From numbers and figures. Samples of math thinking for math lovers . Springer-Verlag 2000, ISBN 3-540-63303-0
2. ^ Robert J. Betts: Using Bonse's Inequality to Find Upper Bounds on Prime Gaps . In: Journal of Integer Sequences . tape 10 , no. 07.3.8 , 2007, ISSN  1530-7638 (English, direct download [PDF; 101 kB ; accessed on March 22, 2020]). Available at Journal of Integer Sequences - Volume 10, 2007. (English).